find d2ydx2 in terms of x and y when y2x3Solutionwe have y2

find d^2y/dx^2 in terms of x and y when y^2=x^3

we have y² = x³

differentiating both sides we get:

2y(dy/dx) = 3x²_..........equation 1

=> dy/dx =x²/2y _..........equation 2

again deffreentiating both sides of equation 1 we get:

2(dy/dx)(dy/dx) + 2y(d²y/dx²) = 6x

=>2(dy/dx)² +2y(d²y/dx²) = 6x

putting the value of dy/dx (which we got from equation 1) in above equation we get:

2(x²/2y)² + 2y(d²y/dx²) = 6x

=> x⁴/2y² + 2y(d²y/dx²) = 6x

or 2y(d²y/dx²) = 6x -( x⁴/2y²)

(d²y/dx²) = [6x -( x⁴/2y²)]/2y

^{(d²y/dx²) = 3(x/y) - (x⁴/4y³)}